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Properties

Label 1620.2.j.b
Level $1620$
Weight $2$
Character orbit 1620.j
Analytic conductor $12.936$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(1133,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.1133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.j (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{25} + 12 q^{37} + 12 q^{55} + 24 q^{61} - 12 q^{67} + 48 q^{85} - 24 q^{91} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1 0 0 0 −2.23225 − 0.130655i 0 −0.792891 + 0.792891i 0 0 0
1133.2 0 0 0 −2.16009 + 0.577937i 0 2.85554 − 2.85554i 0 0 0
1133.3 0 0 0 −1.64124 − 1.51866i 0 −0.689559 + 0.689559i 0 0 0
1133.4 0 0 0 −1.06108 + 1.96828i 0 −3.38622 + 3.38622i 0 0 0
1133.5 0 0 0 −0.144818 − 2.23137i 0 2.46411 − 2.46411i 0 0 0
1133.6 0 0 0 −0.102791 − 2.23370i 0 −0.450982 + 0.450982i 0 0 0
1133.7 0 0 0 0.102791 + 2.23370i 0 −0.450982 + 0.450982i 0 0 0
1133.8 0 0 0 0.144818 + 2.23137i 0 2.46411 − 2.46411i 0 0 0
1133.9 0 0 0 1.06108 − 1.96828i 0 −3.38622 + 3.38622i 0 0 0
1133.10 0 0 0 1.64124 + 1.51866i 0 −0.689559 + 0.689559i 0 0 0
1133.11 0 0 0 2.16009 − 0.577937i 0 2.85554 − 2.85554i 0 0 0
1133.12 0 0 0 2.23225 + 0.130655i 0 −0.792891 + 0.792891i 0 0 0
1457.1 0 0 0 −2.23225 + 0.130655i 0 −0.792891 − 0.792891i 0 0 0
1457.2 0 0 0 −2.16009 − 0.577937i 0 2.85554 + 2.85554i 0 0 0
1457.3 0 0 0 −1.64124 + 1.51866i 0 −0.689559 − 0.689559i 0 0 0
1457.4 0 0 0 −1.06108 − 1.96828i 0 −3.38622 − 3.38622i 0 0 0
1457.5 0 0 0 −0.144818 + 2.23137i 0 2.46411 + 2.46411i 0 0 0
1457.6 0 0 0 −0.102791 + 2.23370i 0 −0.450982 − 0.450982i 0 0 0
1457.7 0 0 0 0.102791 − 2.23370i 0 −0.450982 − 0.450982i 0 0 0
1457.8 0 0 0 0.144818 − 2.23137i 0 2.46411 + 2.46411i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1133.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.j.b 24
3.b odd 2 1 inner 1620.2.j.b 24
5.c odd 4 1 inner 1620.2.j.b 24
9.c even 3 1 180.2.w.a ✓ 24
9.c even 3 1 540.2.x.a 24
9.d odd 6 1 180.2.w.a ✓ 24
9.d odd 6 1 540.2.x.a 24
15.e even 4 1 inner 1620.2.j.b 24
36.f odd 6 1 720.2.cu.d 24
36.h even 6 1 720.2.cu.d 24
45.h odd 6 1 900.2.be.e 24
45.h odd 6 1 2700.2.bf.e 24
45.j even 6 1 900.2.be.e 24
45.j even 6 1 2700.2.bf.e 24
45.k odd 12 1 180.2.w.a ✓ 24
45.k odd 12 1 540.2.x.a 24
45.k odd 12 1 900.2.be.e 24
45.k odd 12 1 2700.2.bf.e 24
45.l even 12 1 180.2.w.a ✓ 24
45.l even 12 1 540.2.x.a 24
45.l even 12 1 900.2.be.e 24
45.l even 12 1 2700.2.bf.e 24
180.v odd 12 1 720.2.cu.d 24
180.x even 12 1 720.2.cu.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.w.a ✓ 24 9.c even 3 1
180.2.w.a ✓ 24 9.d odd 6 1
180.2.w.a ✓ 24 45.k odd 12 1
180.2.w.a ✓ 24 45.l even 12 1
540.2.x.a 24 9.c even 3 1
540.2.x.a 24 9.d odd 6 1
540.2.x.a 24 45.k odd 12 1
540.2.x.a 24 45.l even 12 1
720.2.cu.d 24 36.f odd 6 1
720.2.cu.d 24 36.h even 6 1
720.2.cu.d 24 180.v odd 12 1
720.2.cu.d 24 180.x even 12 1
900.2.be.e 24 45.h odd 6 1
900.2.be.e 24 45.j even 6 1
900.2.be.e 24 45.k odd 12 1
900.2.be.e 24 45.l even 12 1
1620.2.j.b 24 1.a even 1 1 trivial
1620.2.j.b 24 3.b odd 2 1 inner
1620.2.j.b 24 5.c odd 4 1 inner
1620.2.j.b 24 15.e even 4 1 inner
2700.2.bf.e 24 45.h odd 6 1
2700.2.bf.e 24 45.j even 6 1
2700.2.bf.e 24 45.k odd 12 1
2700.2.bf.e 24 45.l even 12 1